Subgroups — #36447597

Order 22784.

454 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26020272	C₂	1	2	yes	maximal
2	C₄×#15667946	C₂	1	1	yes	maximal, characteristic, complemented
3	#26019011	C₂	1	1	yes	maximal, characteristic, complemented
4	#26018154	C₂	1	2	yes	maximal, complemented
5	C₈×#41×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#15667946	—	2	1	no	—
7	#19151622	C₂²	1	2	yes	—
8	C₂×#15667939	—	2	2	no	—
9	C₄×C₄⋊Dic₈₉	C₂²	1	2	yes	—
10	#19151361	—	2	1	no	—
11	#19151357	—	2	2	no	—
12	#19151351	—	2	1	no	—
13	#19151351	—	2	1	no	—
14	C₄×#41×C₈₉	C₂²	1	1	yes	characteristic
15	C₈×Q₈×C₈₉	C₄	1	1	yes	characteristic, complemented
16	C₈×Q₈×C₈₉	C₄	1	1	yes	characteristic, complemented
17	C₈×D₄×C₈₉	—	2	1	no	—
18	C₈×D₄×C₈₉	—	2	1	no	—
19	C₈×D₄×C₈₉	C₄	1	1	yes	characteristic, complemented
20	C₈×D₄×C₈₉	C₄	1	1	yes	characteristic, complemented
21	#431×C₈₉	—	2	1	no	—
22	#431×C₈₉	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₈₉	—	2	1	no	—
24	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
25	C₂×#13083558	—	2	2	no	—
26	C₂×C₄⋊Dic₈₉	—	2	2	no	—
27	C₂×C₄×Dic₈₉	—	2	2	no	—
28	#15667946	—	2	4	no	—
29	C₄⋊#10623494	—	2	2	no	—
30	C₄⋊#10623494	—	2	2	no	—
31	#15667939	—	4	4	no	—
32	#15667919	—	2	2	no	—
33	C₂×#41×C₈₉	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₈₉	—	2	1	no	—
35	C₂²×C₈×C₈₉	—	2	1	no	—
36	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
38	C₄×Q₈×C₈₉	C₂×C₄	1	1	yes	characteristic
39	C₄×Q₈×C₈₉	Q₈	1	1	yes	characteristic
40	C₄×D₄×C₈₉	—	2	1	no	—
41	C₄×D₄×C₈₉	—	2	1	no	—
42	C₄×D₄×C₈₉	C₂×C₄	1	1	yes	characteristic
43	C₄×D₄×C₈₉	Q₈	1	1	yes	characteristic
44	#117×C₈₉	—	2	1	no	—
45	#117×C₈₉	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₈₉	—	2	1	no	—
47	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
48	C₄⋊C₈×C₈₉	—	2	1	no	—
49	C₄⋊C₈×C₈₉	—	2	1	no	—
50	C₄⋊C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
51	C₄⋊C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
52	C₂²⋊C₈×C₈₉	—	2	1	no	—
53	C₂²⋊C₈×C₈₉	—	2	1	no	—
54	C₂²⋊C₈×C₈₉	—	2	1	no	—
55	C₄×C₈×C₈₉	—	2	1	no	—
56	C₄×C₈×C₈₉	—	2	1	no	—
57	C₄×C₈×C₈₉	—	2	1	no	—
58	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
59	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
60	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
61	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
62	C₂²×Dic₈₉	—	4	2	no	—
63	#13083558	—	4	2	no	—
64	#13083558	—	4	2	no	—
65	C₂×#10623494	—	4	2	no	—
66	C₄⋊Dic₈₉	—	2	4	no	—
67	C₄⋊Dic₈₉	—	4	2	no	—
68	C₄×Dic₈₉	—	4	2	no	—
69	C₄×Dic₈₉	—	4	2	no	—
70	#41×C₈₉	—	2	1	no	—
71	#41×C₈₉	—	2	1	no	—
72	#41×C₈₉	D₈	1	2	yes	—
73	#41×C₈₉	Q₁₆	1	2	yes	—
74	C₂×Q₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
75	C₂×D₄×C₈₉	—	2	1	no	—
76	C₂×D₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
77	C₂²×C₄×C₈₉	—	2	1	no	—
78	C₂²×C₄×C₈₉	—	2	1	no	—
79	C₂²×C₄×C₈₉	—	2	1	no	—
80	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
83	C₂×C₈×C₈₉	—	2	2	no	—
84	C₂×C₈×C₈₉	—	2	1	no	—
85	C₂×C₈×C₈₉	—	2	1	no	—
86	C₂×C₈×C₈₉	—	4	1	no	—
87	C₂×C₈×C₈₉	—	4	2	no	—
88	C₂×C₈×C₈₉	—	4	2	no	—
89	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
93	C₄⋊C₄×C₈₉	—	2	1	no	—
94	C₄⋊C₄×C₈₉	—	2	1	no	—
95	C₄⋊C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
96	C₄⋊C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
97	C₂²⋊C₄×C₈₉	—	2	1	no	—
98	C₂²⋊C₄×C₈₉	—	2	1	no	—
99	C₂²⋊C₄×C₈₉	—	2	1	no	—
100	C₄²×C₈₉	—	2	1	no	—
101	C₄²×C₈₉	—	2	1	no	—
102	C₄²×C₈₉	—	2	1	no	—
103	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
104	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
106	C₄²×C₈₉	#41	1	1	yes	characteristic
107	C₂³×C₈₉	—	2	1	no	—
108	C₂³×C₈₉	#121	1	1	yes	characteristic
109	Q₈×C₈₉	—	2	1	no	—
110	Q₈×C₈₉	#107	1	2	yes	complemented
111	D₄×C₈₉	—	2	2	no	—
112	D₄×C₈₉	—	2	2	no	—
113	D₄×C₈₉	—	2	1	no	—
114	D₄×C₈₉	#107	1	2	yes	complemented
115	C₂×C₄×C₈₉	—	2	1	no	—
116	C₂×C₄×C₈₉	—	2	1	no	—
117	C₂×C₄×C₈₉	—	2	1	no	—
118	C₂×C₄×C₈₉	—	2	2	no	—
119	C₂×C₄×C₈₉	—	2	2	no	—
120	C₂×C₄×C₈₉	—	2	1	no	—
121	C₂×C₄×C₈₉	—	2	1	no	—
122	C₂×C₄×C₈₉	—	4	1	no	—
123	C₂×C₄×C₈₉	—	4	2	no	—
124	C₂×C₄×C₈₉	—	4	1	no	—
125	C₂×C₄×C₈₉	—	4	1	no	—
126	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
128	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
129	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
130	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
131	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
132	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
134	C₂×C₄×C₈₉	#135	1	1	yes	characteristic
135	C₂×C₄×C₈₉	#135	1	1	yes	characteristic
136	C₂×Dic₈₉	—	4	4	no	—
137	C₂×Dic₈₉	—	8	2	no	—
138	C₂×Dic₈₉	—	8	2	no	—
139	C₈×C₈₉	—	2	1	no	—
140	C₈×C₈₉	—	2	1	no	—
141	C₈×C₈₉	—	2	1	no	—
142	C₈×C₈₉	—	4	1	no	—
143	C₈×C₈₉	—	4	1	no	—
144	C₈×C₈₉	#104	1	2	yes	complemented
145	#10623494	—	8	2	no	—
146	C₂²×C₈₉	—	2	1	no	—
147	C₂²×C₈₉	—	2	1	no	—
148	C₂²×C₈₉	—	2	2	no	—
149	C₂²×C₈₉	—	4	1	no	—
150	C₂²×C₈₉	—	4	1	no	—
151	C₂²×C₈₉	#380	1	1	yes	characteristic
152	C₂²×C₈₉	#466	1	1	yes	characteristic
153	C₂²×C₈₉	#467	1	1	yes	characteristic
154	C₄×C₈₉	—	2	1	no	—
155	C₄×C₈₉	—	2	2	no	—
156	C₄×C₈₉	—	2	1	no	—
157	C₄×C₈₉	—	2	1	no	—
158	C₄×C₈₉	—	4	1	no	—
159	C₄×C₈₉	—	4	1	no	—
160	C₄×C₈₉	C₂×#104	1	1	yes	characteristic
161	C₄×C₈₉	#421	1	1	yes	characteristic
162	C₄×C₈₉	C₂×#107	1	1	yes	characteristic
163	C₄×C₈₉	#427	1	1	yes	characteristic
164	C₄×C₈₉	#466	1	1	yes	characteristic
165	C₄×C₈₉	#467	1	1	yes	characteristic
166	Dic₈₉	—	8	4	no	—
167	#8667	—	89	1	no	maximal, Sylow
168	C₂×C₈₉	—	2	1	no	—
169	C₂×C₈₉	—	2	1	no	—
170	C₂×C₈₉	—	4	2	no	—
171	C₂×C₈₉	#1806	1	1	yes	characteristic
172	C₂×C₈₉	#1808	1	1	yes	characteristic
173	C₂×C₈₉	#1814	1	1	yes	characteristic
174	C₈×#41	D₈₉	1	1	yes	characteristic
175	#1806	—	89	2	no	—
176	C₄×#107	—	89	1	no	—
177	#1567	—	89	1	no	—
178	#1303	—	89	2	no	—
179	C₈₉	#8667	1	1	yes	characteristic, complemented, Sylow
180	C₄×#41	C₂×D₈₉	1	1	yes	characteristic
181	C₈×Q₈	Dic₈₉	1	1	yes	characteristic, complemented
182	C₈×Q₈	Dic₈₉	1	1	yes	characteristic, complemented
183	C₈×D₄	—	2	1	no	—
184	C₈×D₄	—	2	1	no	—
185	C₈×D₄	Dic₈₉	1	1	yes	characteristic, complemented
186	C₈×D₄	Dic₈₉	1	1	yes	characteristic, complemented
187	#431	—	2	1	no	—
188	#431	C₂×D₈₉	1	1	yes	characteristic
189	C₂×#107	—	178	1	no	—
190	#423	—	89	2	no	—
191	C₂×#104	—	178	2	no	—
192	C₂×C₄×C₈	—	2	1	no	—
193	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
194	C₄×C₄⋊C₄	—	89	2	no	—
195	#340	—	178	1	no	—
196	#339	—	178	2	no	—
197	#335	—	178	1	no	—
198	#335	—	178	1	no	—
199	C₂×#41	D₃₅₆	1	1	yes	characteristic
200	C₂×M₁₆	—	178	2	no	—
201	C₂²×C₈	—	2	1	no	—
202	C₂²×C₈	—	2	1	no	—
203	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
204	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
205	C₄×Q₈	Dic₁₇₈	1	1	yes	characteristic
206	C₄×Q₈	C₂×Dic₈₉	1	1	yes	characteristic
207	C₄×D₄	—	2	1	no	—
208	C₄×D₄	—	2	1	no	—
209	C₄×D₄	Dic₁₇₈	1	1	yes	characteristic
210	C₄×D₄	C₂×Dic₈₉	1	1	yes	characteristic
211	#117	—	2	1	no	—
212	#117	D₃₅₆	1	1	yes	characteristic
213	C₂×C₄⋊C₄	—	178	2	no	—
214	C₂×C₄²	—	2	1	no	—
215	C₂×C₄²	—	178	2	no	—
216	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
217	#107	—	178	4	no	—
218	C₄⋊C₈	—	2	1	no	—
219	C₄⋊C₈	—	2	1	no	—
220	C₄⋊C₈	—	178	2	no	—
221	C₄⋊C₈	—	178	2	no	—
222	C₄⋊C₈	C₂×Dic₈₉	1	1	yes	characteristic
223	C₄⋊C₈	C₂×Dic₈₉	1	1	yes	characteristic
224	#104	—	356	4	no	—
225	C₂²⋊C₈	—	2	1	no	—
226	C₂²⋊C₈	—	2	1	no	—
227	C₂²⋊C₈	—	2	1	no	—
228	C₄×C₈	—	2	1	no	—
229	C₄×C₈	—	2	1	no	—
230	C₄×C₈	—	2	1	no	—
231	C₄×C₈	C₂×Dic₈₉	1	1	yes	characteristic
232	C₄×C₈	C₂×Dic₈₉	1	1	yes	characteristic
233	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
234	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
235	#95	—	178	2	no	—
236	#41	—	2	1	no	—
237	#41	—	2	1	no	—
238	#41	D₇₁₂	1	2	yes	—
239	#41	Dic₃₅₆	1	2	yes	—
240	C₂×Q₈	C₄⋊Dic₈₉	1	1	yes	characteristic
241	C₂×D₄	—	2	1	no	—
242	C₂×D₄	C₄⋊Dic₈₉	1	1	yes	characteristic
243	C₂²×C₄	—	2	1	no	—
244	C₂²×C₄	—	2	1	no	—
245	C₂²×C₄	—	2	1	no	—
246	C₂²×C₄	—	356	2	no	—
247	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
248	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
249	C₂²×C₄	#13083574	1	1	yes	characteristic
250	M₁₆	—	356	2	no	—
251	M₁₆	—	356	2	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	2	1	no	—
254	C₂×C₈	—	2	1	no	—
255	C₂×C₈	—	4	1	no	—
256	C₂×C₈	—	4	2	no	—
257	C₂×C₈	—	4	2	no	—
258	C₂×C₈	—	356	2	no	—
259	C₂×C₈	C₂²⋊Dic₈₉	1	1	yes	characteristic
260	C₂×C₈	C₂²⋊Dic₈₉	1	1	yes	characteristic
261	C₂×C₈	C₂²⋊Dic₈₉	1	1	yes	characteristic
262	C₂×C₈	C₂²⋊Dic₈₉	1	1	yes	characteristic
263	C₄⋊C₄	—	2	1	no	—
264	C₄⋊C₄	—	2	1	no	—
265	C₄⋊C₄	—	356	2	no	—
266	C₄⋊C₄	—	178	4	no	—
267	C₄⋊C₄	C₄⋊Dic₈₉	1	1	yes	characteristic
268	C₄⋊C₄	C₄⋊Dic₈₉	1	1	yes	characteristic
269	C₂²⋊C₄	—	2	1	no	—
270	C₂²⋊C₄	—	2	1	no	—
271	C₂²⋊C₄	—	2	1	no	—
272	C₄²	—	2	1	no	—
273	C₄²	—	2	1	no	—
274	C₄²	—	2	1	no	—
275	C₄²	—	356	2	no	—
276	C₄²	—	356	2	no	—
277	C₄²	C₂²×Dic₈₉	1	1	yes	characteristic
278	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
279	C₄²	C₂×Dic₁₇₈	1	1	yes	characteristic
280	C₄²	#13083574	1	1	yes	characteristic
281	C₂³	—	2	1	no	—
282	C₂³	#15667986	1	1	yes	characteristic
283	Q₈	—	2	1	no	—
284	Q₈	#15667946	1	2	yes	complemented
285	D₄	—	2	2	no	—
286	D₄	—	2	2	no	—
287	D₄	—	2	1	no	—
288	D₄	#15667946	1	2	yes	complemented
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	—	4	1	no	—
300	C₂×C₄	—	356	4	no	—
301	C₂×C₄	—	712	2	no	—
302	C₂×C₄	—	712	2	no	—
303	C₂×C₄	C₂×C₂²⋊Dic₈₉	1	1	yes	characteristic
304	C₂×C₄	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
305	C₂×C₄	#15667971	1	1	yes	characteristic
306	C₂×C₄	#15667986	1	1	yes	characteristic
307	C₂×C₄	#15667991	1	1	yes	characteristic
308	C₂×C₄	#15667991	1	1	yes	characteristic
309	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
310	C₂×C₄	C₂×Dic₃₅₆	1	1	yes	characteristic
311	C₂×C₄	#15668034	1	1	yes	characteristic
312	C₂×C₄	#15668034	1	1	yes	characteristic
313	C₈	—	2	1	no	—
314	C₈	—	2	1	no	—
315	C₈	—	2	1	no	—
316	C₈	—	4	1	no	—
317	C₈	—	4	1	no	—
318	C₈	—	712	2	no	—
319	C₈	#15667939	1	2	yes	complemented
320	C₂²	—	2	1	no	—
321	C₂²	—	2	1	no	—
322	C₂²	—	2	2	no	—
323	C₂²	—	4	1	no	—
324	C₂²	—	4	1	no	—
325	C₂²	#19151458	1	1	yes	characteristic
326	C₂²	#19151834	1	1	yes	characteristic
327	C₂²	#19151839	1	1	yes	characteristic
328	C₄	—	2	1	no	—
329	C₄	—	2	2	no	—
330	C₄	—	2	1	no	—
331	C₄	—	2	1	no	—
332	C₄	—	4	1	no	—
333	C₄	—	4	1	no	—
334	C₄	—	712	4	no	—
335	C₄	C₂×#15667939	1	1	yes	characteristic
336	C₄	#19151615	1	1	yes	characteristic
337	C₄	C₂×#15667946	1	1	yes	characteristic
338	C₄	#19151636	1	1	yes	characteristic
339	C₄	#19151834	1	1	yes	characteristic
340	C₄	#19151839	1	1	yes	characteristic
341	C₂	—	2	1	no	—
342	C₂	—	2	1	no	—
343	C₂	—	4	2	no	—
344	C₂	#26020272	1	1	yes	characteristic
345	C₂	#26020281	1	1	yes	characteristic
346	C₂	#26020306	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#15667986
commutator	C₄²×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈×#41×C₈₉	C₂
socle	C₂²×C₈₉	#380
prime radical	C₂×#41×C₈₉	D₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	C₈×#41	D₈₉
hypocenter	C₈₉	#8667
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₈×#41

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₄×#15667946	C₂	semidirect
3. #26019011	C₂	semidirect
4. #26018154	C₂	semidirect
15. C₈×Q₈×C₈₉	C₄	semidirect
16. C₈×Q₈×C₈₉	C₄	semidirect
19. C₈×D₄×C₈₉	C₄	semidirect
20. C₈×D₄×C₈₉	C₄	semidirect
110. Q₈×C₈₉	#107	semidirect
114. D₄×C₈₉	#107	semidirect
144. C₈×C₈₉	#104	semidirect
179. C₈₉	#8667	semidirect
181. C₈×Q₈	Dic₈₉	semidirect
182. C₈×Q₈	Dic₈₉	semidirect
185. C₈×D₄	Dic₈₉	semidirect
186. C₈×D₄	Dic₈₉	semidirect
284. Q₈	#15667946	semidirect
288. D₄	#15667946	semidirect
319. C₈	#15667939	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#26020272	2
C₄×#15667946	2
#26019011	2
#26018154	2
C₈×#41×C₈₉	2
#8667	89